Frequentist notion of probability

Question

The or rather one of the Frequentist interpretations of probability claims that the statement "The next coin toss )that is executed under conditions X) has a probability of 90% to land heads" simply means: "The next coin toss belongs to an infinite sequence of coin tosses that are executed under conditions X, and which have the limiting relative frequency 90%." Let (x1,x2, .... ) be the infinite sequence of coin tosses under condition X.

My question now is why this should give me any confidence that the next coin that I am interested in (and that is executed under conditions X) will land heads. It seems to me that the 90% are only relevant to my single case, if I make the additional assumption that the next coin toss that I am considering is somehow randomly chosen from this infinite sequence. Or alternatively I have symmetric evidence that the next coin toss be x1 or x2 or x3 or ... .

However I have never read anything similar to that from a defendant of the frequentist notion of probability

My real question is therefore whether:

1. All that (Probability-)Frequentists mean when they claim: "The next coin toss (that is executed under conditions X) has a probability of 90% to land heads" is really: "The next coin toss belongs to an infinite sequence of coin tosses that are executed under conditions X, and which have the limiting relative frequency 90%." or
2. Whether they actually mean something different and I just missunderstand them

Answer 1

It's useful to remember that frequentism was rooted in Machian or positivist radical empiricism. On this radical empiricist picture, all we can talk about are series of observations; we can't observe anything like "real chances" or stochastic causal connections, and so we should avoid that kind of language.

So, yes, #1, that really is all the most rigorous nineteenth century frequentists like Karl Pearson mean by probability.

Later frequentists were less rigorous empiricists, however. For example, Fisher was an indeterminist, and sometimes used stochastic causal connection language. The distribution of coin flip results in X can be described completely by an unobserved parameter rho (which you can interpret as the probability that any one coin flip is heads). For Fisher, rho has a true value, and rho stochastically causes the distribution of coin flip values that we actually observe. By virtue of this causal connection, we can use the percentage of observed flips that are heads to estimate the value of rho. For Karl Pearson, by contrast, this percentage is just a convenient way to summarize the data; since we can't actually observe rho, it's (roughly) meaningless to talk about its "true" value.

So #2 is true for an indeterminist frequentist like Fisher. There are "real chances" and stochastic causation, and we can estimate the true values of unobserved stochastic causes using the properties of long-run tendencies of sequences of observations.